Abstract

We present a method for designing non-absorbing optical diffusers that, when illuminated by a converging beam, produce a specified intensity distribution along the optical axis. To evaluate the performance of the diffusers in imaging systems we calculate the three-dimensional distribution of the mean intensity in the neighborhood of focus. We find that the diffusers can be used as depth-of-focus extenders. We also propose and implement a method of fabricating the designed diffusers on photoresist-coated plates and present some experimental results obtained with the fabricated diffusers.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  19. E. R. Mendez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
    [CrossRef]
  20. E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
    [CrossRef]
  21. E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  25. E. R. M´endez and D. Macias, "Inverse problems in optical scattering," in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer, New York, 2006), pp. 435-465.
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  28. J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176-178 (1969).
    [CrossRef]
  29. E. H. Linfoot and E. Wolf, "Diffraction images in systems with annular apertures," Proc. Phys. Soc. London Sect. B 66, 145149 (1953).
    [CrossRef]

2006

2005

2004

2003

2002

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

2001

1998

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

1996

R. Piestum, B. Spektor and J. Shamir, "Unconventional light distributions in three-dimensional domains," J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

1995

1994

1992

1985

1977

W. T. Welford, "Optical estimation of statistics of surface roughness from light scattering measurements," Opt. Quantum Electron. 9, 269-387 (1977).
[CrossRef]

1969

J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176-178 (1969).
[CrossRef]

1964

1960

1954

1953

E. H. Linfoot and E. Wolf, "Diffraction images in systems with annular apertures," Proc. Phys. Soc. London Sect. B 66, 145149 (1953).
[CrossRef]

Bara, S.

Burvall, A.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, "Axicon - the most important optical element", Optics and Photonics News 16, 34-39 (2005).
[CrossRef]

Campos, J.

Castro, A.

A. Castro and J. Ojeda-Castaneda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474- 3479 (2004).
[CrossRef] [PubMed]

Cathey, W. T.

Chi, W.

Chu, K.

Dainty, J. C.

J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176-178 (1969).
[CrossRef]

Dowski, E. R.

Eliezer, E. B.

Escalera, J. C.

Escamilla, H. M.

Friberg, A. T.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, "Axicon - the most important optical element", Optics and Photonics News 16, 34-39 (2005).
[CrossRef]

Garcia-Guerrero, E. E.

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

E. R. Mendez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
[CrossRef]

George, N.

Gimeno, R.

Gu, Z. H.

Harvey, A. A.

Hernandez-Figueroa, H. E.

Iemmi, C.

Jaroszewicz, Z.

Kolodziejczyk, A.

Kotodziejczyk, A.

Leskova, T. A.

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

E. R. Mendez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
[CrossRef]

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
[CrossRef]

Leyva-Lucero, M.

E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, "Diffraction images in systems with annular apertures," Proc. Phys. Soc. London Sect. B 66, 145149 (1953).
[CrossRef]

Lopez-Coronado, O.

Lu, J. Q.

M´endez, E. R.

Maradudin, A. A.

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

E. R. Mendez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
[CrossRef]

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Mendez, A. A. Maradudin, J. Q. Lu, T. Michel, and M. Nieto-Vesperinas, "Interaction of two optical beams at a symmetric random surface," Appl. Opt. 31, 5878-5889 (1992).
[CrossRef] [PubMed]

E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
[CrossRef]

Marom, E.

McCutchen, C.W.

McLeod, J. H.

Mendez, E. R.

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

Z. H. Gu, H. M. Escamilla, E. R. Mendez, A. A. Maradudin, J. Q. Lu, T. Michel, and M. Nieto-Vesperinas, "Interaction of two optical beams at a symmetric random surface," Appl. Opt. 31, 5878-5889 (1992).
[CrossRef] [PubMed]

E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
[CrossRef]

Mezouari, S.

Michel, T.

Nieto-Vesperinas, M.

Novikov, I. V.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

Piestum, R.

R. Piestum, B. Spektor and J. Shamir, "Unconventional light distributions in three-dimensional domains," J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

Recami, E.

Rosen, J.

Shamir, J.

R. Piestum, B. Spektor and J. Shamir, "Unconventional light distributions in three-dimensional domains," J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

Shchegrov, A. V.

E. R. Mendez, E. E. Garcia-Guerrero, H. M. Escamilla, A. A. Maradudin, T. A. Leskova, and A. V. Shchegrov, "Photofabrication of random achromatic optical diffusers for uniform illumination," Appl. Opt. 40, 1098-1108 (2001).
[CrossRef]

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

Shemer, A.

Sochacki, J.

Spektor, B.

R. Piestum, B. Spektor and J. Shamir, "Unconventional light distributions in three-dimensional domains," J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

Staronski, L. R.

Welford, W. T.

W. T. Welford, "Optical estimation of statistics of surface roughness from light scattering measurements," Opt. Quantum Electron. 9, 269-387 (1977).
[CrossRef]

W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749-753 (1960).
[CrossRef]

Wolf, E.

E. H. Linfoot and E. Wolf, "Diffraction images in systems with annular apertures," Proc. Phys. Soc. London Sect. B 66, 145149 (1953).
[CrossRef]

Yariv, A.

Yzuel, M. J.

Zalevsky, Z.

Zamboni-Rached, M.

Zlotnik, A.

Appl. Opt.

Appl. Phys. Lett.

T. A. Leskova, A. A. Maradudin, I. V. Novikov, A. V. Shchegrov, and E. R. Mendez, "Design of one-dimensional band-limited uniform diffusers of light," Appl. Phys. Lett. 73, 1943-1945 (1998).
[CrossRef]

E. R. Mendez, E. E. Garcia-Guerrero, T. A. Leskova, A. A. Maradudin, J. Mu˜noz-L´opez, and I. Simonsen, "Design of one-dimensional random surfaces with specified scattering properties," Appl. Phys. Lett. 81, 798-800 (2002)
[CrossRef]

J. Mod. Opt.

R. Piestum, B. Spektor and J. Shamir, "Unconventional light distributions in three-dimensional domains," J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J. C. Dainty, "The image of a point for an aberration free lens with a circular pupil," Opt. Commun. 1, 176-178 (1969).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

W. T. Welford, "Optical estimation of statistics of surface roughness from light scattering measurements," Opt. Quantum Electron. 9, 269-387 (1977).
[CrossRef]

Optics and Photonics News

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, "Axicon - the most important optical element", Optics and Photonics News 16, 34-39 (2005).
[CrossRef]

Proc. Phys. Soc. London Sect. B

E. H. Linfoot and E. Wolf, "Diffraction images in systems with annular apertures," Proc. Phys. Soc. London Sect. B 66, 145149 (1953).
[CrossRef]

Other

A. Castro and J. Ojeda-Castaneda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474- 3479 (2004).
[CrossRef] [PubMed]

E. R. M´endez and D. Macias, "Inverse problems in optical scattering," in Light Scattering and Nanoscale Surface Roughness, A. A. Maradudin, ed. (Springer, New York, 2006), pp. 435-465.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, New York, 1992), pp 281-282.

E. R. Mendez, T. A. Leskova, A. A. Maradudin, M. Leyva-Lucero, and J. Munoz-Lopez, "The design of twodimensional random surfaces with specified scattering properties," J. Opt. A 7, S141-S151 (2005).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, UK, 1999), Sect. 8.8.

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Figures (9)

Fig. 1.
Fig. 1.

Diffraction of a converging spherical wave transmitted through the rotationally-symmetric diffuser.

Fig. 2.
Fig. 2.

Intensity distribution produced by the zones with (a) n = 0 and (b) n = 25. The parameters are: a =2cm, R = 15cm, λ = 0.633 μm, N = 100, and αn = 0.

Fig. 3.
Fig. 3.

Calculated mean intensity distribution in the focal region of an optical system with a diffuser designed to produce uniform intensity along the optical axis in the region -2cm ≤ z 0 ≤ 2cm, with a = 2cm, R = 15cm, λ = 0.633μm, and N = 100.

Fig. 4.
Fig. 4.

Axial (a) and transversal (b) normalized intensities corresponding to the mean PSF shown in Fig. 3. The axial response of the system with a clear aperture is too narrow to be shown in (a). The lower (red) curve in (b) represents to the response of the system with the clear aperture.

Fig. 5.
Fig. 5.

Illustration of the focusing behavior of different zones of the diffuser. The probability of focusing on a point of the optical axis is uniform within the specified region and zero outside.

Fig. 6.
Fig. 6.

Transverse intensity distribution obtained with a single realization of a diffuser whose mean intensity is shown in Fig. 3. In (a) and (b) the wavelength λ = 0.633μm, while in (c) and (d) the results represent an average over the intensities obtained for 400 equally spaced wavelengths in the region 450μm to 650μm. In (a) and (c) z 0 = -1cm and in (b) and (d) z 0 = 1cm. The pictures show a region of 100×100μm in the (x 0,y 0)-plane.

Fig. 7.
Fig. 7.

Schematic diagram of the setup employed for the fabrication of the circularly symmetric random diffusers.

Fig. 8.
Fig. 8.

Realization of a random surface profile H(r) with 100 zones (a), and mask generated from it (b).

Fig. 9.
Fig. 9.

Images of a wheel-like binary object under white light illumination in a well-corrected 4f imaging system with a = 0.5cm and f = 30cm. (a) Clear-aperture system in the best-focus plane (z 0 = 0). (b) System with diffuser in the best-focus plane. (c) Clear-aperture system in an out-of-focus plane (z 0 = 8mm). (d) System with diffuser in the same out-of-focus plane. The images shows a region of about 4×4mm of the object.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

ψ z 0 r 0 = i k 0 A 0 R 2 e i k 0 z 0 0 a t ( r ) J 0 ( k 0 r 0 R r ) exp { i k 0 z 0 r 2 2 R 2 } rdr .
t ( r ) = exp { i v 3 H ( r ) } ,
ψ z 0 0 = C 0 0 a exp { i v 3 H ( r ) } exp { i κ z 0 r 2 } rdr ,
t = r 2 ,
ψ z 0 0 = 1 2 C 0 0 a 2 dt exp { i v 3 h ( t ) } exp { i κ z 0 t } ,
h ( t ) = H ( t ) .
I z 0 0 = ψ z 0 0 2
= 1 4 C 0 2 0 a 2 dt 0 a 2 dt′ exp { i v 3 [ h ( t ) h ( t′ ) ] } exp { i κ z 0 ( t t′ ) } ,
I z 0 0 = 1 4 C 0 2 0 a 2 dt 0 a 2 dt′ exp [ i κ z 0 ( t t′ ) ] exp [ i v 3 ( t t′ ) h′ ( t′ ) ] ,
h ( t ) = α n b t + β n , n b 2 t ( n + 1 ) b 2 , n = 0,1,2 , . . . , N 1 .
f ( γ ) = δ ( γ α n )
β n + 1 = β n ( n + 1 ) ( α n + 1 α n ) b
β n = [ α 0 + α 1 + + α n 1 n α n ] b , n 1 ,
0 a 2 d t 0 a 2 dt′ exp [ i κ z 0 ( t t′ ) ] exp [ i v 3 ( t t′ ) h′ ( t′ ) ]
= 0 a 2 dt n = 0 N 1 n b 2 ( n + 1 ) b 2 dt′ exp [ i κ z 0 ( t t′ ) ] exp [ i v 3 ( t t ) ( α n b ) ]
= 0 a 2 d t n = 0 N 1 n b 2 ( n + 1 ) b 2 d t exp [ i κ z 0 ( t t ) ] d γ f ( γ ) [ i v 3 ( t t ) ( γ b ) ] ,
dγf ( γ ) 0 a 2 dt exp { i [ κ z 0 + ( v 3 γ b ) ] t } 2
= 4 dγf ( γ ) sin 2 [ κ z 0 + ( v 3 γ b ) ] a 2 2 [ κ z 0 + ( v 3 γ b ) ] 2 .
I z 0 0 = C 0 2 a 4 4 dγf ( γ ) sin c 2 ( κ a 2 2 z 0 + v 3 a 2 γ 2 b ) ,
sin c 2 A x π A δ ( x ) ,
I z 0 0 = π a 2 C 0 2 b 2 v 3 f ( κ b v 3 z 0 ) .
f ( γ ) = 1 π a 2 1 C 0 2 2 v 3 b I ( v 3 κ b γ , 0 ) .
H ( r ) = α n b r 2 + β n , n b r n + 1 b , n = 0,1,2 , . . . , N 1 .
I z 0 0 = I 0 θ ( z m z 0 ) ,
f ( γ ) = 2 π v 3 I 0 a 2 b C 0 2 θ ( κ b v 3 z m γ ) .
I 0 = π 4 a 2 C 0 2 κ z m .
I z 0 0 = π 4 a 2 C 0 2 κ z m θ ( z m z 0 )
f ( γ ) = v 3 2 b 1 κ z m θ ( κ b v 3 z m γ ) .
ψ z 0 r 0 = C 0 0 a J 0 ( k 0 r 0 R r ) e i v 3 H ( r ) i κ r 2 z 0 rdr .
ψ z 0 r 0 = C 0 n = 0 N 1 e i v 3 β n ψ n ( z 0 , r 0 ; α n ) ,
ψ n ( z 0 , r 0 ; α n ) = n b n + 1 b rdr J 0 ( k 0 r 0 R r ) exp { i ( κ z 0 + v 3 α n b ) r 2 } .
I z 0 r 0 = ψ z 0 r 0 2
= C 0 2 m = 0 N 1 n = 0 N 1 e i v 3 ( β m β n ) ψ m ( z 0 , r 0 ; α m ) ψ n * ( z 0 , r 0 ; α n ) .
I z 0 r 0 = C 0 2 n = 0 N 1 ψ n ( z 0 , z 0 ; α n ) 2 .
ψ n ( z 0 , 0 ; α n ) = n b n + 1 b rdr exp { i ( κ z 0 + v 3 α n b ) r 2 } .
ψ n ( z 0 , 0 ; α n ) = b 2 2 exp { i ( κ z 0 + v 3 α n b ) b 2 ( n + 1 2 ) } sin c [ ( κ z 0 + v 3 α n b ) b 2 2 ] ,
I z 0 0 = C 0 2 α 4 4 N sin c 2 [ 1 N ( k 0 a 2 4 R 2 z 0 + v 3 a 2 2 b α n ) ] ,
I z 0 0 = C 0 2 a 4 4 sin c 2 ( k 0 a 2 4 R 2 z 0 + v 3 a 2 2 b α n ) .
I z 0 r 0 = v 3 2 b C 0 2 κ z m n = 0 N 1 κ b v 3 z m κ b v 3 z m d γ ψ n ( z 0 , r 0 ; γ ) 2 .

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